Approximation of the Fractional Schrodinger Propagator on Compact Manifolds

Let L be a second order positive,elliptic differential operator that is self-adjoint with respect to some C^(∞)density dx on a compact connected manifold M.We proved that if 0<α<1,α/2<s<αand f∈H^(s)(M)then the fractional Schrodinger propagator e^(itLα/2) on M satisfies eitLα/2 f(x)−f(x)=o(t^(s/α−ε))almost everywhere as t→0^(+),for anyε>0.